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Middle School Archive

Browse Middle School Algebra
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Direct and indirect variation.
    Inequalities and negative numbers.
    Positive/negative integer rules.
    Solving simple linear equations.

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Algebra as a Metaphor for Life [06/11/2002]

I've been given 'proofs' that -1 = 1, and that 2 > 4. Can you show me why they aren't true?

Algebra: Average of an Average [10/20/1997]

If the one department hires 50 people/month, and another 80, what is the average number of people/month hired?

Algebra (Equation) [12/12/1997]

What is the answer to the problem: 15-(5m-4) = 34?

Algebra: Factoring Fractions [07/30/1997]

Question: 8ab/4a2-b2 divided by 16b2/8a2-4ab.

Algebra for Equivalence [11/12/2011]

A teacher's warning about algebraic manipulations troubles a student. With simple examples, Doctor Peterson shows how dividing by a variable may destroy solutions, how taking roots may introduce extraneous ones, and how factoring makes both more explicit, all while emphasizing caution with assumptions.

Algebraic Explanation for Wrist Multiplication [09/10/2004]

The trick behind the technique of wrist-multiplying is explained by use of a standard algebraic multiplication formula.

Algebraic Fractions [5/27/1996]

How would you solve for x and find the least common denominator in the equation 2/x = 4/3?

Algebraic Sentences [01/30/2002]

Is there a trick to understanding algebraic sentences?

The Algebra of Complements and Supplements [01/25/1999]

What are complements and supplements? How do you translate these concepts into algebra?

Algebra Problem [8/7/1996]

How do I solve for T? 2t-12t+18 = 0

Algebra Refresher [12/31/1997]

I am trying to figure out how to solve this equation: (x^3 + 3x^2 - 4x + 3) / (x+1).

Algebra - Solving Equations [11/03/1997]

I am having a hard time solving some of my algebra problems...

Algebra Terms: 'Evaluate' and 'Simplify' [7/30/1996]

What do the terms "evaluate" and "simplify" mean in expressions like "evaluate 3-{2-3(2-7)}" or "simplify -3x{7-(3x-5)}"?

Algebra Test Questions [1/25/1996]

1. Find 3 consecutive integers such that the sum of the first and third is 84. 2. The length of a rectangle is 5 less than twice the width, the perimeter of the rectangle is 80; find the dimensions of the rectangle. 3. Solve these equations: c + 2/5 = 3/7 and 1/6 = a + 3/a - 4.

Alternate Solution Path to Venn Diagram Problem [05/12/2009]

There are 30 students in a math class. Twelve belong to the computer club, and eight belong to the photography club. Three belong to both clubs. How many belong to neither club?

Angles of a Parallelogram [8/24/1996]

In a parallelogram ABCD, angle ABC = 2 * angle BCD; what is ADC?

Another Order of Operations Problem [6/19/1995]

(1+1*8-9*4) / (8*9+7*2) = ?

Applying the Distributive Property to Division or Subtraction [12/29/2006]

The distributive property is easily shown with multiplication and addition. Can it be used with subtraction or division?

Are (-a)^3 and -a^3 the Same Thing? [11/08/2005]

One question on a test I took was (-a) * (-a) * (-a) = ?. I answered (-a)^3. My teacher said the answer is -a^3 (same answer without the parantheses). Aren't the two answers the same?

Area of an Inner Rectangle [01/19/2003]

Fred and Wilma's house had a 40 x 72 ft rectangular yard. Wilma cultivated 5/12 of the area in flowers, which she grew in an even border around the central grass area. Every morning Fred walked Dino around the central grass area inside the flowers. How many laps did he have to do to walk at least 1/4 mile?

Asking Price of a House [10/14/2001]

If a seller wants to clear $50,000 for his house and the real estate commission is 6%, what should be the asking price for the house?

Assigning Variables for Age [02/21/2003]

John is 32. He is twice as old as Jim was when John was as old as Jim is now. How old is Jim?

Associative Property [5/22/1996]

What is the associative property in algebra?

Associative Property of Multiplication and Grouping Symbols [10/22/2006]

Does the Associative Property of Multiplication make grouping symbols unnecessary, since (a x b) x c = a x (b x c) = a x b x c?

Average Speed [04/29/2001]

A test track is 1 mile around. A driver drives 1 lap at 160mph. How fast must he drive the second lap in order to average 200mph for the two laps?

Average Speed of a Caterpillar [09/08/99]

A caterpillar crawls at 21 in/hr to a vegetable patch and returns over exactly the same distance at 7 in/hr. What its average speed over the entire journey?

Avoiding Careless Mistakes [01/30/2002]

How can I avoid careless mistakes?

Ball Thrown between Trains [10/15/2003]

Two trains A and B each of length 100m travel in opposite directions in parallel tracks. The speeds are 20m/s and 30m/s respectively. A boy sitting in the front end of train A throws a ball to a boy sitting in the front end of train B when they are at the closest distance. The speed of the ball is 2m/s. The ball, instead of reaching the boy, hits the rear end of the train. Find the distance between the parallel tracks.

Baseballs, Buckets, and Milk Cartons [10/09/2001]

The mass of a baseball is 50g. What is the mass of a bucket when 2 buckets = 6 blocks; 1 bucket + 1 block = 2 milk cartons; and 2 baseballs = 1 milk carton.

Baseball Wins [9/10/1996]

A baseball team played 18 games and won 4 more games than it lost. How many games did it win?

Basic Algebra Problem [6/24/1996]

Can I solve a2-10 = ?

Bedbugs and Beds [09/14/1997]

In each single bed you can find 7 bedbugs, and in each double bed 13 bedbugs. If there are 106 bedbugs in all, how many double beds are there?

Beginning Algebra [02/13/2003]

How come, for x + 3 = -2, x = -5?

Billions Taste the Rainbow Thousands of Billions of Billions of Times [03/06/2013]

A student who knows facts about populations and physical chemistry struggles to determine how long would it take the world to consume a mole of Skittles. Doctor Ian shows the power of algebra by introducing a key assumption that makes this Fermi problem approachable.

Birthday Number Puzzle Variations [11/18/2002]

A friend showed me a number trick that works no matter what number I start with. How does it work? He also said that the year 2002 is the only year it will ever work. Is that true?

Boat Crossing River, Lake [8/3/1996]

It takes a motorboat one hr. to go downstream from A to B and 1.5 hrs. to go up-stream from B to A...

Borrowing and Returning Books [03/26/2001]

Four students borrowed books from a bookshelf and eight students returned books. There are 27 books on the shelf. How many books were there in the beginning?

Break-Even Algebra [10/24/2014]

A teen struggles to interpret a word problem into an algebraic inequality. By relating inequalities to equalities and applying some dimensional analysis, Doctor Ian steps through the question.

Burning Candles [06/10/1999]

A candle is lit at 4:30 and burns until 10:30, and a shorter one is lit at 6:00 and goes out at 10:00. If they were the same length at 8:30, what were their original lengths?

Buying Stamps [08/21/1997]

You buy .02 and .15 stamps, paying $1.56 in all. There are 10 more .02 than .15 stamps; how many of each kind did you buy?

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